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%%文档的题目、作者与日期
%\author{王立庆（2020级数学与应用数学1班） }
\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\title{常微分方程期中练习二}
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\date{2023 年 11 月 7 日}
%\date{March 9, 2021}

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\begin{document}

\maketitle

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\begin{enumerate}

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\item  %第1题
求解微分方程 $ \frac{dy}{dx}=\frac{y}{2x-y^2}$. 

\vspace{0.2cm}


\vspace{5cm}

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\item  %第2题
求解微分方程的初值问题 $ \frac{dy}{dx}=\frac{\ln x}{1+y^2}, \,\, y(1)=3$.  


\vspace{0.2cm}


\vspace{5cm}

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\item  %第3题
判断微分方程 $\frac{y}{x}dx + (y^3+\ln x)dy = 0$ 是否为恰当方程，如果是恰当方程请求解。

\vspace{0.2cm}


\vspace{5cm}

\newpage
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\item  %第4题
判断函数 $f(x,y)=xy^2$ 在所给区域上是否满足利普希茨条件。
\begin{enumerate}
\item  $a\le x\le b, c\le y\le d$. 
\item  $a\le x\le b, -\infty< y< \infty$. 
\end{enumerate}

\vspace{0.2cm}


\vspace{5cm}

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\item  %第5题
%记 $p=\frac{dy}{dx}$, 求隐式微分方程 $y(y-2xp)^2=2p$ 的奇解。
已知微分方程的解函数族为 $xy=cy-c^2$, 求其包络以及对应的微分方程的奇解。

\vspace{0.2cm}


\vspace{7cm}

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\item  %第6题
设有一根粗细和密度都均匀的木杆，一端固定，从水平状态自由摆动。
设木杆的质量为 $m$, 长度为 $L$. 
求木杆摆到竖直状态所需要的时间。

\vspace{0.2cm}


\vspace{5cm}

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\item  %第7题
求解微分方程的初值问题 $x\frac{dy}{dx} - 2y = \cos(x), y(\pi)=0$. 

\vspace{0.2cm}


\vspace{5cm}

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\item  %第8题
%求微分方程 $y'' + xy' + x^2y = \sin(x)$ 的所有解的最大存在区间。
设函数 $f(x,y)$ 与 $\frac{\partial f}{\partial y}(x,y)$ 在 $Oxy$ 平面上连续。
使用解的皮卡存在唯一性定理与解的延伸定理，求微分方程初值问题
$\frac{dy}{dx} = (y^2-1)f(x,y), \,\, y(0)=0$
的解的存在区间。

\vspace{0.2cm}


\vspace{6cm}

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\item  %第9题
有一个跑步的人沿着以原点为圆心，半径为30米的圆周跑步，恒定速率为每秒2米。 
有一条狗从原点出发，以每秒3米的恒定速率跑向跑步者，方向始终指向跑步者。
求出狗的运动轨迹。


\vspace{0.2cm}


\vspace{5cm}


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\item  %第10题
求解关于未知函数 $y(x)$ 的方程，$$y(x) = 1 + x^2 + 2\int_0^x y(t)dt. $$

\vspace{0.2cm}


\vspace{5cm}

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\item  %第11题
记 $p=\frac{dy}{dx}$, 构造克莱罗方程 $y=xp+f(p)$, 使得 $y=\sin(x)$ 是其奇解。


\vspace{0.2cm}


\vspace{6cm}

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\item  %第12题
求解常微分方程组 
\begin{eqnarray*}
\left\{\begin{array}{rcl}
\frac{dx}{dt} &=& 4x - xy, \\
\frac{dy}{dt} &=& -9y + 3xy.
\end{array}\right. 
\end{eqnarray*}


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\end{enumerate}


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\end{document}

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